A (mathematically) sound investment strategy

It is common wisdom in the investment community that a long-term investor saving for his future would do well to invest in high-risk/high-return assets when he is young, slowly switching his portfolio over to low-risk/low-return assets as he grows older. I have never seen a mathematical demonstration of this, and would be interested in finding one.

This isn't obviously a mathematical question, but since econ.stackexchange.com doesn't exist yet, I'm asking it here.

We might begin to formulate the question along the following lines: for each year $t=0,1,\dots,T-1$ of his life an investor chooses to put a portion of his total wealth $W_t$ into either a risk-free bond, earning interest at rate $r$, or into a risky asset whose return is a random variable with mean $\mu>r$ and variance $\sigma^2$ (we could say it's a normal random variable, for simplicity). Let's say that at time $t$ he invests a fraction $\phi_t$ in the risky asset and $1-\phi_t$ in the riskless bond.

He is also able to invest an additional amount $P_t$ into his portfolio at time $t$ (which comes, for example, from his salary). We could take $P_t=P$ deterministically to begin with, and later generalize to non-homogeneous or stochastic $P_t$.

Then the question becomes: for a given level of risk-taking, what strategy $\{\phi_t\}$ maximises his expected wealth at time $T$? As proxies for the expected wealth and level of risk, we could take $E(W_T)$ and $\mathrm{Var}(W_T)$. A common approach is to introduce the Lagrange multiplier $\lambda$ and solve the unconstrained optimisation problem

$$\max_{\phi} \, E(W_T) - \lambda\mathrm{Var}(W_T)$$

to create an `optimal frontier' of strategies in $(E(W_T), \mathrm{Var}(W_T))$ space, although this isn't necessarily the best strategy for solving. I haven't taken this idea much further than this, and would welcome any comments or suggestions.

I have now given this some additional thought, and added my progress in an answer below. This is not complete though, and I'd welcome further input!

Solution 1:

I thought about this problem a little today, and have made some progress. This is not a full answer, but may be of interest anyway.

I think a more sensible approach is to take a continuum approximation. To avoid confusion I will take $S_t$ to be the amount of money you have in your portfolio at time $t$, with $\phi_t$ invested in an asset paying a risky return with mean $\mu$ and volatility $\sigma$, and $1-\phi_t$ invested in a riskless bond paying return $r$. I also assume that you pay an amount $p_t$ per unit time into the portfolio. Then the portfolio process $S_t$ satisfies

$$dS_t = p_t dt + (1-\phi_t) r S_t dt + \phi_t \mu S_t dt + \phi_t \sigma dW_t$$

where $W_t$ is a standard Brownian motion, and the terms represent money paid in, interest from the bond, return from the stock and volatility from the stock (the risky term). Defining

$$\Phi_{s,t} = \exp \left( \int_s^t (r + (\mu-r)\phi_u + \tfrac{1}{2} \sigma^2 \phi_u^2) du + \int_s^t \sigma \phi_u dW_u \right)$$

we can write the solution for the value of the portfolio at time $t$ as

$$S_t = S_0\Phi_{0,t} + \int_0^t p_s \Phi_{s,t} ds $$

The question is now how to calculate the expectation and variance of such a portfolio, as a functional of the asset allocation process $\phi_t$, and then formulate and solve an Euler-Lagrange equation for $\phi_t$.